Volume 8 (2007), Issue 1, Article 15, 21 pp.emis.maths.adelaide.edu.au/journals/JIPAM/images/059_06... · 2007-04-02 · Volume 8 (2007), Issue 1, Article 15, 21 pp. THE EQUAL VARIABLE

Volume 8 (2007), Issue 1, Article 15, 21 pp.

THE EQUAL VARIABLE METHOD

VASILE CÎRTOAJE

DEPARTMENT OFAUTOMATION AND COMPUTERS

UNIVERSITY OF PLOIESTI

BUCURESTI39, ROMANIA

[email protected]

Received 01 March, 2006; accepted 17 April, 2006Communicated by P.S. Bullen

ABSTRACT. The Equal Variable Method (called alson−1 Equal Variable Method on the Math-links Site - Inequalities Forum) can be used to prove some difficult symmetric inequalities in-volving either three power means or, more general, two power means and an expression of formf(x1) + f(x2) + · · ·+ f(xn).

Key words and phrases:Symmetric inequalities, Power means, EV-Theorem.

2000Mathematics Subject Classification.26D10, 26D20.

1. STATEMENT OF RESULTS

In order to state and prove the Equal Variable Theorem (EV-Theorem) we require the follow-ing lemma and proposition.

Lemma 1.1. Let a, b, c be fixed non-negative real numbers, not all equal and at most one ofthem equal to zero, and letx ≤ y ≤ z be non-negative real numbers such that

x + y + z = a + b + c, xp + yp + zp = ap + bp + cp,

wherep ∈ (−∞, 0] ∪ (1,∞). For p = 0, the second equation isxyz = abc > 0. Then, thereexist two non-negative real numbersx1 andx2 with x1 < x2 such thatx ∈ [x1, x2]. Moreover,

(1) if x = x1 andp ≤ 0, then0 < x < y = z;(2) if x = x1 andp > 1, then either0 = x < y ≤ z or 0 < x < y = z;(3) if x ∈ (x1, x2), thenx < y < z;(4) if x = x2, thenx = y < z.

Proposition 1.2. Let a, b, c be fixed non-negative real numbers, not all equal and at most oneof them equal to zero, and let0 ≤ x ≤ y ≤ z such that

x + y + z = a + b + c, xp + yp + zp = ap + bp + cp,

059-06

mailto:[email protected]

http://www.ams.org/msc/

2 VASILE CÎRTOAJE

wherep ∈ (−∞, 0] ∪ (1,∞). For p = 0, the second equation isxyz = abc > 0. Letf(u) be a

differentiable function on(0,∞), such thatg(x) = f ′(x

1p−1

)is strictly convex on(0,∞), and

let

F3(x, y, z) = f(x) + f(y) + f(z).

(1) If p ≤ 0, thenF3 is maximal only for0 < x = y < z, and is minimal only for0 < x < y = z;

(2) If p > 1 and eitherf(u) is continuous atu = 0 or limu→0

f(u) = −∞, thenF3 is maximal

only for0 < x = y < z, and is minimal only for eitherx = 0 or 0 < x < y = z.

Theorem 1.3 (Equal Variable Theorem (EV-Theorem)). Let a1, a2, . . . , an (n ≥ 3) be fixednon-negative real numbers, and let0 ≤ x1 ≤ x2 ≤ · · · ≤ xn such that

x1 + x2 + · · ·+ xn = a1 + a2 + · · ·+ an,

xp1 + xp

2 + · · ·+ xpn = ap

1 + ap2 + · · ·+ ap

n,

wherep is a real number,p 6= 1. For p = 0, the second equation isx1x2 · · ·xn = a1a2 · · · an >0. Letf(u) be a differentiable function on(0,∞) such that

g(x) = f ′(x

1p−1

)is strictly convex on(0,∞), and let

Fn(x1, x2, . . . , xn) = f(x1) + f(x2) + · · ·+ f(xn).

(1) If p ≤ 0, thenFn is maximal for0 < x1 = x2 = · · · = xn−1 ≤ xn, and is minimal for0 < x1 ≤ x2 = x3 = · · · = xn;

(2) If p > 0 and eitherf(u) is continuous atu = 0 or limu→0

f(u) = −∞, thenFn is

maximal for0 ≤ x1 = x2 = · · · = xn−1 ≤ xn, and is minimal for eitherx1 = 0 or0 < x1 ≤ x2 = x3 = · · · = xn.

Remark 1.4. Let 0 < α < β. If the functionf is differentiable on(α, β) and the function

g(x) = f ′(x

1p−1

)is strictly convex on(αp−1, βp−1) or (βp−1, αp−1), then the EV-Theorem

holds true forx1, x2, . . . , xn ∈ (α, β).

By Theorem 1.3, we easily obtain some particular results, which are very useful in applica-tions.

Corollary 1.5. Let a1, a2, . . . , an (n ≥ 3) be fixed non-negative numbers, and let0 ≤ x1 ≤x2 ≤ · · · ≤ xn such that

x1 + x2 + · · ·+ xn = a1 + a2 + · · ·+ an,

x21 + x2

2 + · · ·+ x2n = a2

1 + a22 + · · ·+ a2

n.

Letf be a differentiable function on(0,∞) such thatg(x) = f ′(x) is strictly convex on(0,∞).Moreover, eitherf(x) is continuous atx = 0 or lim

x→0f(x) = −∞. Then,

Fn = f(x1) + f(x2) + · · ·+ f(xn)

is maximal for0 ≤ x1 = x2 = · · · = xn−1 ≤ xn, and is minimal for eitherx1 = 0 or0 < x1 ≤ x2 = x3 = · · · = xn.

J. Inequal. Pure and Appl. Math., 8(1) (2007), Art. 15, 21 pp. http://jipam.vu.edu.au/

http://jipam.vu.edu.au/

EQUAL VARIABLE METHOD 3

Corollary 1.6. Let a1, a2, . . . , an (n ≥ 3) be fixed positive numbers, and let0 < x1 ≤ x2 ≤· · · ≤ xn such that

x1 + x2 + · · ·+ xn = a1 + a2 + · · ·+ an,

1

x1

+1

x2

+ · · ·+ 1

xn

=1

a1

+1

a2

+ · · ·+ 1

an

.

Let f be a differentiable function on(0,∞) such thatg(x) = f ′(

1√x

)is strictly convex on

(0,∞). Then,Fn = f(x1) + f(x2) + · · ·+ f(xn)

is maximal for0 < x1 = x2 = · · · = xn−1 ≤ xn, and is minimal for0 < x1 ≤ x2 = x3 = · · · =xn.

Corollary 1.7. Let a1, a2, . . . , an (n ≥ 3) be fixed positive numbers, and let0 < x1 ≤ x2 ≤· · · ≤ xn such that

x1 + x2 + · · ·+ xn = a1 + a2 + · · ·+ an, x1x2 · · ·xn = a1a1 · · · an.

Letf be a differentiable function on(0,∞) such thatg(x) = f ′(

1x

)is strictly convex on(0,∞).

Then,Fn = f(x1) + f(x2) + · · ·+ f(xn)

is maximal for0 < x1 = x2 = · · · = xn−1 ≤ xn, and is minimal for0 < x1 ≤ x2 = x3 = · · · =xn.

Corollary 1.8. Let a1, a2, . . . , an (n ≥ 3) be fixed non-negative numbers, and let0 ≤ x1 ≤x2 ≤ · · · ≤ xn such that

x1 + x2 + · · ·+ xn = a1 + a2 + · · ·+ an,

xp1 + xp

2 + · · ·+ xpn = ap

1 + ap2 + · · ·+ ap

n,

wherep is a real number,p 6= 0 andp 6= 1.

(a) For p < 0, P = x1x2 · · ·xn is minimal when0 < x1 = x2 = · · · = xn−1 ≤ xn, and ismaximal when0 < x1 ≤ x2 = x3 = · · · = xn.

(b) For p > 0, P = x1x2 · · ·xn is maximal when0 ≤ x1 = x2 = · · · = xn−1 ≤ xn, and isminimal when eitherx1 = 0 or 0 < x1 ≤ x2 = x3 = · · · = xn.

Corollary 1.9. Let a1, a2, . . . , an (n ≥ 3) be fixed non-negative numbers, let0 ≤ x1 ≤ x2 ≤· · · ≤ xn such that

x1 + x2 + · · ·+ xn = a1 + a2 + · · ·+ an,

xp1 + xp

2 + · · ·+ xpn = ap

1 + ap2 + · · ·+ ap

n,

and letE = xq1 + xq

2 + · · ·+ xqn.

Case 1.p ≤ 0 (p = 0 yieldsx1x2 · · ·xn = a1a2 · · · an > 0).

(a) For q ∈ (p, 0) ∪ (1,∞), E is maximal when0 < x1 = x2 = · · · = xn−1 ≤ xn, and isminimal when0 < x1 ≤ x2 = x3 = · · · = xn.

(b) For q ∈ (−∞, p) ∪ (0, 1), E is minimal when0 < x1 = x2 = · · · = xn−1 ≤ xn, and ismaximal when0 < x1 ≤ x2 = x3 = · · · = xn.

Case 2.0 < p < 1.

(a) For q ∈ (0, p) ∪ (1,∞), E is maximal when0 ≤ x1 = x2 = · · · = xn−1 ≤ xn, and isminimal when eitherx1 = 0 or 0 < x1 ≤ x2 = x3 = · · · = xn.



4 VASILE CÎRTOAJE

(b) For q ∈ (−∞, 0) ∪ (p, 1), E is minimal when0 ≤ x1 = x2 = · · · = xn−1 ≤ xn, and ismaximal when eitherx1 = 0 or 0 < x1 ≤ x2 = x3 = · · · = xn.

Case 3.p > 1.

(a) For q ∈ (0, 1) ∪ (p,∞), E is maximal when0 ≤ x1 = x2 = · · · = xn−1 ≤ xn, and isminimal when eitherx1 = 0 or 0 < x1 ≤ x2 = x3 = · · · = xn.

(b) For q ∈ (−∞, 0) ∪ (1, p), E is minimal when0 ≤ x1 = x2 = · · · = xn−1 ≤ xn, and ismaximal when eitherx1 = 0 or 0 < x1 ≤ x2 = x3 = · · · = xn.

2. PROOFS

Proof of Lemma 1.1.Let a ≤ b ≤ c. Note that in the excluded casesa = b = c anda = b = 0,there is a single triple(x, y, z) which verifies the conditions

x + y + z = a + b + c and xp + yp + zp = ap + bp + cp.

Consider now three cases:p = 0, p < 0 andp > 1.

A. Casep = 0 (xyz = abc > 0). LetS = a+b+c3

andP = 3√

abc, whereS > P > 0 by AM-GMInequality. We have

x + y + z = 3S, xyz = P 3,

and from0 < x ≤ y ≤ z andx < z, it follows that0 < x < P . Now let

f = y + z − 2√

yz.

It is clear thatf ≥ 0, with equality if and only ify = z. Writing f as a function ofx,

f(x) = 3S − x− 2P

√P

x,

we have

f ′(x) =P

x

√P

x− 1 > 0,

and hence the functionf(x) is strictly increasing. Sincef(P ) = 3(S − P ) > 0, the equationf(x) = 0 has a unique positive rootx1, 0 < x1 < P . Fromf(x) ≥ 0, it follows thatx ≥ x1.

Sub-casex = x1. Sincef(x) = f(x1) = 0 andf = 0 impliesy = z, we have0 < x < y = z.

Sub-casex > x1. We havef(x) > 0 andy < z. Consider now thaty andz depend onx. Fromx + y(x) + z(x) = 3S andx · y(x) · z(x) = P 3, we get1 + y′ + z′ = 0 and 1

x+ y′

y+ z′

z= 0.

Hence,

y′(x) =y(x− z)

x(z − y), z′(x) =

z(y − x)

x(z − y).

Sincey′(x) < 0, the functiony(x) is strictly decreasing. Sincey(x1) > x1 (see sub-casex = x1), there existsx2 > x1 such thaty(x2) = x2, y(x) > x for x1 < x < x2 andy(x) < xfor x > x2. Taking into account thaty ≥ x, it follows thatx1 < x ≤ x2. On the other hand, wesee thatz′(x) > 0 for x1 < x < x2. Consequently, the functionz(x) is strictly increasing, andhencez(x) > z(x1) = y(x1) > y(x). Finally, we conclude thatx < y < z for x ∈ (x1, x2),andx = y < z for x = x2.

B. Casep < 0. DenoteS = a+b+c3

andR =(

ap+bp+cp

3

) 1p . Taking into account that

x + y + z = 3S, xp + yp + zp = 3Rp,




from 0 < x ≤ y ≤ z andx < z we getx < S and31p R < x < R. Let

h = (y + z)

(yp + zp

2

)−1p

− 2.

By the AM-GM Inequality, we have

h ≥ 2√

yz1√

yz− 2 = 0,

with equality if and only ify = z. Writing nowh as a function ofx,

h(x) = (3S − x)

(3Rp − xp

2

)−1p

− 2,

from

h′(x) =3Rp

2

(3Rp − xp

2

)−1−pp

[(S

x

)(R

x

)−p

− 1

]> 0

it follows that h(x) is strictly increasing. Sinceh(x) ≥ 0 andh(3

1p R)

= −2, the equation

h(x) = 0 has a unique rootx1 andx ≥ x1 > 31p R.

Sub-casex = x1. Sincef(x) = f(x1) = 0, andf = 0 impliesy = z, we have0 < x < y = z.

Sub-casex > x1. We haveh(x) > 0 andy < z. Consider now thaty andz depend onx.From x + y(x) + z(x) = 3S andxp + y(x)p + z(x)p = 3Rp, we get1 + y′ + z′ = 0 andxp−1 + yp−1y′ + zp−1z′ = 0, and hence

y′(x) =xp−1 − zp−1

zp−1 − yp−1, z′(x) =

xp−1 − yp−1

yp−1 − zp−1.

Sincey′(x) > 0, the functiony(x) is strictly decreasing. Sincey(x1) > x1 (see sub-casex = x1), there existsx2 > x1 such thaty(x2) = x2, y(x) > x for x1 < x < x2, andy(x) < x forx > x2. The conditiony ≥ x yieldsx1 < x ≤ x2. We see now thatz′(x) > 0 for x1 < x < x2.Consequently, the functionz(x) is strictly increasing, and hencez(x) > z(x1) = y(x1) > y(x).Finally, we havex < y < z for x ∈ (x1, x2) andx = y < z for x = x2.

C. Casep > 1. DenotingS = a+b+c3

andR =(

ap+bp+cp

3

) 1p yields

x + y + z = 3S, xp + yp + zp = 3Rp.

By Jensen’s inequality applied to the convex functiong(u) = up, we haveR > S, and hencex < S < R. Let

h =2

y + z

(yp + zp

2

) 1p

− 1.

By Jensen’s Inequality, we geth ≥ 0, with equality if only if y = z. From

h(x) =2

3S − x

(3Rp − xp

2

) 1p

− 1

and

h′(x) =3

(3S − x)2

(3Rp − xp

2

) 1−pp

(Rp − Sxp−1) > 0,

it follows that the functionh(x) is strictly increasing, andh(x) ≥ 0 impliesx ≥ x1. In the caseh(0) ≥ 0 we havex1 = 0, and in the caseh(0) < 0 we havex1 > 0 andh(x1) = 0.



6 VASILE CÎRTOAJE

Sub-casex = x1. If h(0) ≥ 0, then0 = x1 < y(x1) ≤ z(x1). If h(0) < 0, thenh(x1) = 0, andsinceh = 0 impliesy = z, we have0 < x1 < y(x1) = z(x1).

Sub-casex > x1. Sinceh(x) is strictly increasing, forx > x1 we haveh(x) > h(x1) ≥ 0,henceh(x) > 0 andy < z. Fromx + y(x) + z(x) = 3S andxp + yp(x) + zp(x) = 3Rp, we get

y′(x) =xp−1 − zp−1

zp−1 − yp−1, z′(x) =

yp−1 − xp−1

zp−1 − yp−1.

Sincey′(x) < 0, the functiony(x) is strictly decreasing. Taking account ofy(x1) > x1 (seesub-casex = x1), there existsx2 > x1 such thaty(x2) = x2, y(x) > x for x1 < x < x2,andy(x) < x for x > x2. The conditiony ≥ x implies x1 < x ≤ x2. We see now thatz′(x) > 0 for x1 < x < x2. Consequently, the functionz(x) is strictly increasing, and hencez(x) > z(x1) ≥ y(x1) > y(x). Finally, we conclude thatx < y < z for x ∈ (x1, x2), andx = y < z for x = x2. �

Proof of Proposition 1.2.Consider the function

F (x) = f(x) + f(y(x)) + f(z(x))

defined onx ∈ [x1, x2]. We claim thatF (x) is minimal forx = x1 and is maximal forx = x2.If this assertion is true, then by Lemma 1.1 it follows that:

(a) F (x) is minimal for 0 < x = y < z in the casep ≤ 0, or for eitherx = 0 or0 < x < y = z in the casep > 1;

(b) F (x) is maximal for0 < x = y < z.

In order to prove the claim, assume thatx ∈ (x1, x2). By Lemma 1.1, we have0 < x < y <z. From

x + y(x) + z(x) = a + b + c and

xp + yp(x) + zp(x) = ap + bp + cp,

we gety′ + z′ = −1, yp−1y′ + zp−1z′ = −xp−1,

whence

y′ =xp−1 − zp−1

zp−1 − yp−1, z′ =

xp−1 − yp−1

yp−1 − zp−1.

It is easy to check that this result is also valid forp = 0. We have

F ′(x) = f ′(x) + y′f ′(y) + z′f ′(z)

and

F ′(x)

(xp−1 − yp−1)(xp−1 − zp−1)

=g(xp−1)

(xp−1 − yp−1)(xp−1 − zp−1)+

g(yp−1)

(yp−1 − zp−1)(yp−1 − xp−1)

+g(zp−1)

(zp−1 − xp−1)(zp−1 − yp−1).

Sinceg is strictly convex, the right hand side is positive. On the other hand,

(xp−1 − yp−1)(xp−1 − zp−1) > 0.

These results implyF ′(x) > 0. Consequently, the functionF (x) is strictly increasing forx ∈ (x1, x2). Excepting the trivial case whenp > 1, x1 = 0 and lim

u→0f(u) = −∞, the function




F (x) is continuous on[x1, x2], and hence is minimal only forx = x1, and is maximal only forx = x2. �

Proof of Theorem 1.3.We will consider two cases.

Casep ∈ (−∞, 0]∪(1,∞). Excepting the trivial case whenp > 1, x1 = 0 andlimu→0

f(u) = −∞,

the functionFn(x1, x2, . . . , xn) attains its minimum and maximum values, and the conclusionfollows from Proposition 1.2 above, via contradiction. For example, let us consider the casep ≤0. In order to prove thatFn is maximal for0 < x1 = x2 = · · · = xn−1 ≤ xn, we assume, for thesake of contradiction, thatFn attains its maximum at(b1, b2, . . . , bn) with b1 ≤ b2 ≤ · · · ≤ bn

andb1 < bn−1. Letx1, xn−1, xn be positive numbers such thatx1 + xn−1 + xn = b1 + bn−1 + bn

andxp1 + xp

n−1 + xpn = bp

1 + bpn−1 + bp

n. According to Proposition 1.2, the expression

F3(x1, xn−1, xn) = f(x1) + f(xn−1) + f(xn)

is maximal only forx1 = xn−1 < xn, which contradicts the assumption thatFn attains itsmaximum at(b1, b2, . . . , bn) with b1 < bn−1.

Casep ∈ (0, 1). This case reduces to the casep > 1, replacing each of theai by a1p

i , each of

thexi by x1p

i , and thenp by 1p. Thus, we obtain the sufficient condition thath(x) = xf ′

(x

11−p

)to be strictly convex on(0,∞). We claim that this condition is equivalent to the condition that

g(x) = f ′(x

1p−1

)to be strictly convex on(0,∞). Actually, for our proof, it suffices to show

that if g(x) is strictly convex on(0,∞), thenh(x) is strictly convex on(0,∞). To show this,we see thatg

(1x

)= 1

xh(x). Sinceg(x) is strictly convex on(0,∞), by Jensen’s inequality we

have

ug

(1

x

)+ vg

(1

y

)> (u + v)g

( ux

+ vy

u + v

)for anyx, y, u, v > 0 with x 6= y. This inequality is equivalent to

u

xh(x) +

v

yh(y) >

(u

x+

v

y

)h

(u + vux

+ vy

).

Substitutingu = tx andv = (1− t)y, wheret ∈ (0, 1), reduces the inequality to

th(x) + (1− t)h(y) > h(tx + (1− t)y),

which shows us thath(x) is strictly convex on(0,∞). �

Proof of Corollary 1.8.We will apply Theorem 1.3 to the functionf(u) = p ln u. We see thatlimu→0

f(u) = −∞ for p > 0, and

f ′(u) =p

u, g(x) = f ′

(x

1p−1

)= px

11−p , g′′(x) =

p2

(1− p)2x

2p−11−p .

Sinceg′′(x) > 0 for x > 0, the functiong(x) is strictly convex on(0,∞), and the conclusionfollows by Theorem 1.3. �

Proof of Corollary 1.9.We will apply Theorem 1.3 to the function

f(u) = q(q − 1)(q − p)uq.

For p > 0, it is easy to check that eitherf(u) is continuous atu = 0 (in the caseq > 0) orlimu→0

f(u) = −∞ (in the caseq < 0). We have

f ′(u) = q2(q − 1)(q − p)uq−1



8 VASILE CÎRTOAJE

and

g(x) = f ′(x

1p−1

)= q2(q − 1)(q − p)x

q−1p−1 ,

g′′(x) =q2(q − 1)2(q − p)2

(p− 1)2x

2p−11−p .

Sinceg′′(x) > 0 for x > 0, the functiong(x) is strictly convex on(0,∞), and the conclusionfollows by Theorem 1.3. �

3. APPLICATIONS

Proposition 3.1.Letx, y, z be non-negative real numbers such thatx+y+z = 2. If r0 ≤ r ≤ 3,wherer0 = ln 2

ln 3−ln 2≈ 1.71, then

xr(y + z) + yr(z + x) + zr(x + y) ≤ 2.

Proof. Rewrite the inequality in the homogeneous form

xr+1 + yr+1 + zr+1 + 2

(x + y + z

2

)r+1

≥ (x + y + z)(xr + yr + zr),

and apply Corollary 1.9 (casep = r andq = r + 1):If 0 ≤ x ≤ y ≤ z such that

x + y + z = constant and

xr + yr + zr = constant,

then the sumxr+1 + yr+1 + zr+1 is minimal when eitherx = 0 or 0 < x ≤ y = z.Casex = 0. The initial inequality becomes

yz(yr−1 + zr−1) ≤ 2,

wherey + z = 2. Since0 < r − 1 ≤ 2, by the Power Mean inequality we have

yr−1 + zr−1

2≤(

y2 + z2

2

) r−12

.

Thus, it suffices to show that

yz

(y2 + z2

2

) r−12

≤ 1.

Taking account ofy2 + z2

2=

2(y2 + z2)

(y + z)2≥ 1 and

r − 1

2≤ 1,

we have

1− yz

(y2 + z2

2

) r−12

≥ 1− yz

(y2 + z2

2

)=

(y + z)4

16− yz(y2 + z2)

2

=(y − z)4

16≥ 0.

Case0 < x ≤ y = z. In the homogeneous inequality we may leave aside the constraintx + y + z = 2, and considery = z = 1, 0 < x ≤ 1. The inequality reduces to(

1 +x

2

)r+1

− xr − x− 1 ≥ 0.




Since(1 + x

2

)r+1is increasing andxr is decreasing in respect tor, it suffices to considerr = r0.

Let

f(x) =(1 +

x

2

)r0+1

− xr0 − x− 1.

We have

f ′(x) =r0 + 1

2

(1 +

x

2

)r0

− r0xr0−1 − 1,

1

r0

f ′′(x) =r0 + 1

4

(1 +

x

2

)r0

− r0 − 1

x2−r0.

Sincef ′′(x) is strictly increasing on(0, 1], f ′′(0+) = −∞ and

1

r0

f ′′(1) =r0 + 1

4

(3

2

)r0

− r0 + 1

=r0 + 1

2− r0 + 1 =

3− r0

2> 0,

there existsx1 ∈ (0, 1) such thatf ′′(x1) = 0, f ′′(x) < 0 for x ∈ (0, x1), andf ′′(x) > 0 forx ∈ (x1, 1]. Therefore, the functionf ′(x) is strictly decreasing forx ∈ [0, x1], and strictlyincreasing forx ∈ [x1, 1]. Since

f ′(0) =r0 − 1

2> 0 and f ′(1) =

r0 + 1

2

[(3

2

)r0

− 2

]= 0,

there existsx2 ∈ (0, x1) such thatf ′(x2) = 0, f ′(x) > 0 for x ∈ [0, x2), andf ′(x) < 0 forx ∈ (x2, 1). Thus, the functionf(x) is strictly increasing forx ∈ [0, x2], and strictly decreasingfor x ∈ [x2, 1]. Sincef(0) = f(1) = 0, it follows thatf(x) ≥ 0 for 0 < x ≤ 1, establishing thedesired result.

For x ≤ y ≤ z, equality occurs whenx = 0 andy = z = 1. Moreover, forr = r0, equalityholds again whenx = y = z = 1. �

Proposition 3.2([12]). Let x, y, z be non-negative real numbers such thatxy + yz + zx = 3.If 1 < r ≤ 2, then

xr(y + z) + yr(z + x) + zr(x + y) ≥ 6.

Proof. Rewrite the inequality in the homogeneous form

xr(y + z) + yr(z + x) + zr(x + y) ≥ 6

(xy + yz + zx

3

) r+12

.

For convenience, we may leave aside the constraintxy+yz+zx = 3. Using now the constraintx + y + z = 1, the inequality becomes

xr(1− x) + yr(1− y) + zr(1− z) ≥ 6

(1− x2 − y2 − z2

6

) r+12

.

To prove it, we will apply Corollary 1.5 to the functionf(u) = −ur(1− u) for 0 ≤ u ≤ 1. Wehavef ′(u) = −rur−1 + (r + 1)ur and

g(x) = f ′(x) = −rxr−1 + (r + 1)xr, g′′(x) = r(r − 1)xr−3[(r + 1)x + 2− r].

Sinceg′′(x) > 0 for x > 0, g(x) is strictly convex on[0,∞). According to Corollary 1.5,if 0 ≤ x ≤ y ≤ z such thatx + y + z = 1 andx2 + y2 + z2 = constant, then the sumf(x) + f(y) + f(z) is maximal for0 ≤ x = y ≤ z.



10 VASILE CÎRTOAJE

Thus, we have only to prove the original inequality in the casex = y ≤ z. This means, toprove that0 < x ≤ 1 ≤ y andx2 + 2xz = 3 implies

xr(x + z) + xzr ≥ 3.

Let f(x) = xr(x + z) + xzr − 3, with z = 3−x2

2x.

Differentiating the equationx2 + 2xz = 3 yieldsz′ = −(x+z)x

. Then,

f ′(x) = (r + 1)xr + rxr−1z + zr + (xr + rxzr−1)z′

= (xr−1 − zr−1)[rx + (r − 1)z] ≤ 0.

The functionf(x) is strictly decreasing on[0, 1], and hencef(x) ≥ f(1) = 0 for 0 < x ≤ 1.Equality occurs if and only ifx = y = z = 1. �

Proposition 3.3([5]). If x1, x2, . . . , xn are positive real numbers such that

x1 + x2 + · · ·+ xn =1

x1

+1

x2

+ · · ·+ 1

xn

,

then1

1 + (n− 1)x1

+1

1 + (n− 1)x2

+ · · ·+ 1

1 + (n− 1)xn

≥ 1.

Proof. We have to consider two cases.

Casen = 2. The inequality is verified as equality.

Casen ≥ 3. Assume that0 < x1 ≤ x2 ≤ · · · ≤ xn, and then apply Corollary 1.6 to the functionf(u) = 1

1+(n−1)ufor u > 0. We havef ′(u) = −(n−1)

[1+(n−1)u]2and

g(x) = f ′(

1√x

)=

−(n− 1)x

(√

x + n− 1)2 ,

g′′(x) =3(n− 1)2

2√

x (√

x + n− 1)4 .

Sinceg′′(x) > 0, g(x) is strictly convex on(0,∞). According to Corollary 1.6, if0 < x1 ≤x2 ≤ · · · ≤ xn such that

x1 + x2 + · · ·+ xn = constant and

1

x1

+1

x2

+ · · ·+ 1

xn

= constant,

then the sumf(x1) + f(x2) + · · ·+ f(xn) is minimal when0 < x1 ≤ x2 = x3 = · · · = xn.Thus, we have to prove the inequality

1

1 + (n− 1)x+

n− 1

1 + (n− 1)y≥ 1,

under the constraints0 < x ≤ 1 ≤ y and

x + (n− 1)y =1

x+

n− 1

y.

The last constraint is equivalent to

(n− 1)(y − 1) =y(1− x2)

x(1 + y).




Since1

1 + (n− 1)x+

n− 1

1 + (n− 1)y− 1

=1

1 + (n− 1)x− 1

n+

n− 1

1 + (n− 1)y− n− 1

n

=(n− 1)(1− x)

n[1 + (n− 1)x]− (n− 1)2(y − 1)

n[1 + (n− 1)y]

=(n− 1)(1− x)

n[1 + (n− 1)x]− (n− 1)y(1− x2)

nx(1 + y)[1 + (n− 1)y],

we must show that

x(1 + y)[1 + (n− 1)y] ≥ y(1 + x)[1 + (n− 1)x],

which reduces to(y − x)[(n− 1)xy − 1] ≥ 0.

Sincey − x ≥ 0, we have still to prove that

(n− 1)xy ≥ 1.

Indeed, fromx + (n− 1)y = 1x

+ n−1y

we getxy = y+(n−1)xx+(n−1)y

, and hence

(n− 1)xy − 1 =n(n− 2)x

x + (n− 1)y> 0.

Forn ≥ 3, one has equality if and only ifx1 = x2 = · · · = xn = 1. �

Proposition 3.4([10]). Leta1, a2, . . . , an be positive real numbers such thata1a2 · · · an = 1. Ifm is a positive integer satisfyingm ≥ n− 1, then

am1 + am

2 + · · ·+ amn + (m− 1)n ≥ m

(1

a1

+1

a2

+ · · ·+ 1

an

).

Proof. Forn = 2 (hencem ≥ 1), the inequality reduces to

am1 + am

2 + 2m− 2 ≥ m(a1 + a2).

We can prove it by summing the inequalitiesam1 ≥ 1+m(a1−1) andam

2 ≥ 1+m(a2−1), whichare straightforward consequences of Bernoulli’s inequality. Forn ≥ 3, replacinga1, a2, . . . , an

by 1x1

, 1x2

, . . . , 1xn

, respectively, we have to show that

1

xm1

+1

xm2

+ · · ·+ 1

xmn

+ (m− 1)n ≥ m(x1 + x2 + · · ·+ xn)

for x1x2 · · ·xn = 1. Assume0 < x1 ≤ x2 ≤ · · · ≤ xn and apply Corollary 1.9 (casep = 0 andq = −m):

If 0 < x1 ≤ x2 ≤ · · · ≤ xn such that


x1x2 · · ·xn = 1,

then the sum1xm1

+ 1xm2

+ · · ·+ 1xm

nis minimal when0 < x1 = x2 = · · · = xn−1 ≤ xn.

Thus, it suffices to prove the inequality forx1 = x2 = · · · = xn−1 = x ≤ 1, xn = y andxn−1y = 1, when it reduces to:

n− 1

xm+

1

ym+ (m− 1)n ≥ m(n− 1)x + my.



12 VASILE CÎRTOAJE

By the AM-GM inequality, we have

n− 1

xm+ (m− n + 1) ≥ m

xn−1= my.

Then, we have still to show that

1

ym− 1 ≥ m(n− 1)(x− 1).

This inequality is equivalent to

xmn−m − 1−m(n− 1)(x− 1) ≥ 0

and(x− 1)[(xmn−m−1 − 1) + (xmn−m−2 − 1) + · · ·+ (x− 1)] ≥ 0.

The last inequality is clearly true. Forn = 2 andm = 1, the inequality becomes equality.Otherwise, equality occurs if and only ifa1 = a2 = · · · = an = 1. �

Proposition 3.5([6]). Letx1, x2, . . . , xn be non-negative real numbers such thatx1+x2+ · · ·+xn = n. If k is a positive integer satisfying2 ≤ k ≤ n + 2, andr =

(n

n−1

)k−1 − 1, then

xk1 + xk

2 + · · ·+ xkn − n ≥ nr(1− x1x2 · · ·xn).

Proof. If n = 2, then the inequality reduces toxk1 + xk

2 − 2 ≥ (2k − 2)x1x2. For k = 2 andk = 3, this inequality becomes equality, while fork = 4 it reduces to6x1x2(1 − x1x2) ≥ 0,which is clearly true.

Consider nown ≥ 3 and 0 ≤ x1 ≤ x2 ≤ · · · ≤ xn. Towards proving the inequality,we will apply Corollary 1.8 (casep = k > 0): If 0 ≤ x1 ≤ x2 ≤ · · · ≤ xn such thatx1 + x2 + · · · + xn = n andxk

1 + xk2 + · · · + xk

n = constant, then the productx1x2 · · ·xn isminimal when eitherx1 = 0 or 0 < x1 ≤ x2 = x3 = · · · = xn.

Casex1 = 0. The inequality reduces to

xk2 + · · ·+ xk

n ≥nk

(n− 1)k−1,

with x2 + · · · + xn = n, This inequality follows by applying Jensen’s inequality to the convexfunctionf(u) = uk:

xk2 + · · ·+ xk

n ≥ (n− 1)

(x2 + · · ·+ xn

n− 1

)k

.

Case0 < x1 ≤ x2 = x3 = · · · = xn. Denotingx1 = x andx2 = x3 = · · · = xn = y, we haveto prove that for0 < x ≤ 1 ≤ y andx + (n− 1)y = n, the inequality holds:

xk + (n− 1)yk + nrxyn−1 − n(r + 1) ≥ 0.

Write the inequality asf(x) ≥ 0, where

f(x) = xk + (n− 1)yk + nrxyn−1 − n(r + 1), with y =n− x

n− 1.

We see thatf(0) = f(1) = 0. Sincey′ = −1n−1

, we have

f ′(x) = k(xk−1 − yk−1) + nryn−2(y − x)

= (y − x)[nryn−2 − k(yk−2 + yk−3x + · · ·+ xk−2)]

= (y − x)yn−2[nr − kg(x)],




where

g(x) =1

yn−k+

x

yn−k+1+ · · ·+ xk−2

yn−2.

Since the functiony(x) = n−xn−1

is strictly decreasing, the functiong(x) is strictly increasing for2 ≤ k ≤ n. Fork = n + 1, we have

g(x) = y + x +x2

y+ · · ·+ xn−1

yn−2

=(n− 2)x + n

n− 1+

x2

y+ · · ·+ xn−1

yn−2,

and fork = n + 2, we have

g(x) = y2 + yx + x2 +x3

y+ · · ·+ xn

yn−2

=(n2 − 3n + 3)x2 + n(n− 3)x + n2

(n− 1)2+

x3

y+ · · ·+ xn

yn−2.

Therefore, the functiong(x) is strictly increasing for2 ≤ k ≤ n + 2, and the function

h(x) = nr − kg(x)

is strictly decreasing. Note that

f ′(x) = (y − x)yn−2h(x).

We assert thath(0) > 0 andh(1) < 0. If our claim is true, then there existsx1 ∈ (0, 1) such thath(x1) = 0, h(x) > 0 for x ∈ [0, x1), andh(x) < 0 for x ∈ (x1, 1]. Consequently,f(x) is strictlyincreasing forx ∈ [0, x1], and strictly decreasing forx ∈ [x1, 1]. Sincef(0) = f(1) = 0, itfollows thatf(x) ≥ 0 for 0 < x ≤ 1, and the proof is completed.

In order to prove thath(0) > 0, we assume thath(0) ≤ 0. Then,h(x) < 0 for x ∈ (0, 1),f ′(x) < 0 for x ∈ (0, 1), andf(x) is strictly decreasing forx ∈ [0, 1], which contradictsf(0) = f(1). Also, if h(1) ≥ 0, thenh(x) > 0 for x ∈ (0, 1), f ′(x) > 0 for x ∈ (0, 1), andf(x) is strictly increasing forx ∈ [0, 1], which also contradictsf(0) = f(1).

Forn ≥ 3 andx1 ≤ x2 ≤ · · · ≤ xn, equality occurs whenx1 = x2 = · · · = xn = 1, and alsowhenx1 = 0 andx2 = · · · = xn = n

n−1. �

Remark 3.6. Fork = 2, k = 3 andk = 4, we get the following nice inequalities:

(n− 1)(x21 + x2

2 + · · ·+ x2n) + nx1x2 · · ·xn ≥ n2,

(n− 1)2(x31 + x3

2 + · · ·+ x3n) + n(2n− 1)x1x2 · · ·xn ≥ n3,

(n− 1)3(x41 + x4

2 + · · ·+ x4n) + n(3n2 − 3n + 1)x1x2 · · ·xn ≥ n4.

Remark 3.7. The inequality fork = n was posted in 2004 on the Mathlinks Site - InequalitiesForum by Gabriel Dospinescu and Calin Popa.

Proposition 3.8([11]). Letx1, x2, . . . , xn be positive real numbers such that1x1

+ 1x2

+· · ·+ 1xn

=n. Then

x1 + x2 + · · ·+ xn − n ≤ en−1(x1x2 · · ·xn − 1),

whereen−1 =(1 + 1

n−1

)n−1< e.



14 VASILE CÎRTOAJE

Proof. Replacing each of thexi by 1ai

, the statement becomes as follows:If a1, a2, . . . , an are positive numbers such thata1 + a2 + · · ·+ an = n, then

a1a2 · · · an

(1

a1

+1

a2

+ · · ·+ 1

an

− n + en−1

)≤ en−1.

It is easy to check that the inequality holds forn = 2. Consider nown ≥ 3, assume that0 < a1 ≤ a2 ≤ · · · ≤ an and apply Corollary 1.8 (casep = −1): If 0 < a1 ≤ a2 ≤ · · · ≤ an

such thata1 +a2 + · · ·+an = n and 1a1

+ 1a2

+ · · ·+ 1an

= constant, then the producta1a2 · · · an

is maximal when0 < a1 ≤ a2 = a3 = · · · = an.Denotinga1 = x anda2 = a3 = · · · = an = y, we have to prove that for0 < x ≤ 1 ≤ y <

nn−1

andx + (n− 1)y = n, the inequality holds:

yn−1 + (n− 1)xyn−2 − (n− en−1)xyn−1 ≤ en−1.

Letting

f(x) = yn−1 + (n− 1)xyn−2 − (n− en−1)xyn−1 − en−1, with

y =n− x

n− 1,

we must show thatf(x) ≤ 0 for 0 < x ≤ 1. We see thatf(0) = f(1) = 0. Sincey′ = −1n−1

, wehave

f ′(x)

yn−3= (y − x)[n− 2− (n− en−1)y] = (y − x)h(x),

whereh(x) = n− 2− (n− en−1)

n− x

n− 1is a linear increasing function.

Let us show thath(0) < 0 andh(1) > 0. If h(0) ≥ 0, thenh(x) > 0 for x ∈ (0, 1),hencef ′(x) > 0 for x ∈ (0, 1), andf(x) is strictly increasing forx ∈ [0, 1], which contradictsf(0) = f(1). Also,h(1) = en−1 − 2 > 0.

Fromh(0) < 0 andh(1) > 0, it follows that there existsx1 ∈ (0, 1) such thath(x1) = 0,h(x) < 0 for x ∈ [0, x1), andh(x) > 0 for x ∈ (x1, 1]. Consequently,f(x) is strictly decreasingfor x ∈ [0, x1], and strictly increasing forx ∈ [x1, 1]. Sincef(0) = f(1) = 0, it follows thatf(x) ≤ 0 for 0 ≤ x ≤ 1.

Forn ≥ 3, equality occurs whenx1 = x2 = · · · = xn = 1. �

Proposition 3.9([9]). If x1, x2, . . . , xn are positive real numbers, then

xn1 + xn

2 + · · ·+ xnn + n(n− 1)x1x2 · · ·xn

≥ x1x2 · · ·xn(x1 + x2 + · · ·+ xn)

(1

x1

+1

x2

+ · · ·+ 1

xn

).

Proof. For n = 2, one has equality. Assume now thatn ≥ 3, 0 < x1 ≤ x2 ≤ · · · ≤ xn andapply Corollary 1.9 (casep = 0): If 0 < x1 ≤ x2 ≤ · · · ≤ xn such that


x1x2 · · ·xn = constant,

then the sumxn1 + xn

2 + · · ·+ xnn is minimal and the sum1

x1+ 1

x2+ · · ·+ 1

xnis maximal when

0 < x1 ≤ x2 = x3 = · · · = xn.Thus, it suffices to prove the inequality for0 < x1 ≤ 1 andx2 = x3 = · · · = xn = 1. The

inequality becomesxn

1 + (n− 2)x1 ≥ (n− 1)x21,




and is equivalent to

x1(x1 − 1)[(xn−21 − 1) + (xn−3

1 − 1) + · · ·+ (x1 − 1)] ≥ 0,

which is clearly true. Forn ≥ 3, equality occurs if and only ifx1 = x2 = · · · = xn. �

Proposition 3.10([14]). If x1, x2, . . . , xn are non-negative real numbers, then

(n− 1)(xn1 + xn

2 + · · ·+ xnn) + nx1x2 · · ·xn

≥ (x1 + x2 + · · ·+ xn)(xn−11 + xn−1

2 + · · ·+ xn−1n ).

Proof. For n = 2, one has equality. Forn ≥ 3, assume that0 ≤ x1 ≤ x2 ≤ · · · ≤ xn

and apply Corollary 1.9 (casep = n and q = n − 1) and Corollary 1.8 (casep = n): If0 ≤ x1 ≤ x2 ≤ · · · ≤ xn such that


xn1 + xn

2 + · · ·+ xnn = constant,

then the sumxn−11 +xn−1

2 + · · ·+xn−1n is maximal and the productx1x2 · · ·xn is minimal when

eitherx1 = 0 or 0 < x1 ≤ x2 = x3 = · · · = xn.So, it suffices to consider the casesx1 = 0 and0 < x1 ≤ x2 = x3 = · · · = xn.

Casex1 = 0. The inequality reduces to

(n− 1)(xn2 + · · ·+ xn

n) ≥ (x2 + · · ·+ xn)(xn−12 + · · ·+ xn−1

n ),

which immediately follows by Chebyshev’s inequality.Case0 < x1 ≤ x2 = x3 = · · · = xn. Settingx2 = x3 = · · · = xn = 1, the inequality reducesto:

(n− 2)xn1 + x1 ≥ (n− 1)xn−1

1 .

Rewriting this inequality as

x1(x1 − 1)[xn−31 (x1 − 1) + xn−4

1 (x21 − 1) + · · ·+ (xn−2

1 − 1)] ≥ 0,

we see that it is clearly true. Forn ≥ 3 andx1 ≤ x2 ≤ · · · ≤ xn equality occurs whenx1 = x2 = · · · = xn, and forx1 = 0 andx2 = · · · = xn. �

Proposition 3.11([8]). If x1, x2, . . . , xn are positive real numbers, then

(x1 + x2 + · · ·+ xn − n)

(1

x1

+1

x2

+ · · ·+ 1

xn

− n

)+ x1x2 · · ·xn +

1

x1x2 · · ·xn

≥ 2.

Proof. Forn = 2, the inequality reduces to

(1− x1)2(1− x2)

2

x1x2

≥ 0.

For n ≥ 3, assume that0 < x1 ≤ x2 ≤ · · · ≤ xn. Since the inequality preserves its formby replacing each numberxi with 1

xi, we may considerx1x2 · · ·xn ≥ 1. So, by the AM-GM

inequality we get

x1 + x2 + · · ·+ xn − n ≥ n n√

x1x2 · · ·xn − n ≥ 0,

and we may apply Corollary 1.9 (casep = 0 andq = −1): If 0 ≤ x1 ≤ x2 ≤ · · · ≤ xn suchthat


x1x2 · · ·xn = constant,

then the sum1x1

+ 1x2

+ · · ·+ 1xn

is minimal when0 < x1 = x2 = · · · = xn−1 ≤ xn.



16 VASILE CÎRTOAJE

According to this statement, it suffices to considerx1 = x2 = · · · = xn−1 = x andxn = y,when the inequality reduces to

((n− 1)x + y − n)

(n− 1

x+

1

y− n

)+ xn−1y +

1

xn−1y≥ 2,

or (xn−1 +

n− 1

x− n

)y +

[1

xn−1+ (n− 1)x− n

]1

y≥ n(n− 1)(x− 1)2

x.

Since

xn−1 +n− 1

x− n =

x− 1

x[(xn−1 − 1) + (xn−2 − 1) + · · ·+ (x− 1)]

=(x− 1)2

x[xn−2 + 2xn−3 + · · ·+ (n− 1)]

and1

xn−1+ (n− 1)x− n =

(x− 1)2

x

[1

xn−2+

2

xn−3+ · · ·+ (n− 1)

],

it is enough to show that

[xn−2 + 2xn−3 + · · ·+ (n− 1)]y +

[1

xn−2+

2

xn−3+ · · ·+ (n− 1)

]1

y≥ n(n− 1).

This inequality is equivalent to(xn−2y +

1

xn−2y− 2

)+ 2

(xn−3y +

1

xn−3y− 2

)+ · · ·+ (n− 1)

(y +

1

y− 2

)≥ 0,

or(xn−2y − 1)2

xn−2y+

2(xn−3y − 1)2

xn−3y+ · · ·+ (n− 1)(y − 1)2

y≥ 0,

which is clearly true. Equality occurs if and only ifn− 1 of the numbersxi are equal to 1. �

Proposition 3.12([15]). If x1, x2, . . . , xn are non-negative real numbers such thatx1 + x2 +· · ·+ xn = n, then

(x1x2 · · ·xn)1√

n−1 (x21 + x2

2 + · · ·+ x2n) ≤ n.

Proof. For n = 2, the inequality reduces to2(x1x2 − 1)2 ≥ 0. For n ≥ 3, assume that0 ≤ x1 ≤ x2 ≤ · · · ≤ xn and apply Corollary 1.8 (casep = 2): If 0 ≤ x1 ≤ x2 ≤ · · · ≤ xn

such thatx1 +x2 + · · ·+xn = n andx21 +x2

2 + · · ·+x2n = constant, then the productx1x2 · · ·xn

is maximal when0 ≤ x1 = x2 = · · · = xn−1 ≤ xn.Consequently, it suffices to show that the inequality holds forx1 = x2 = · · · = xn−1 = x and

xn = y, where0 ≤ x ≤ 1 ≤ y and(n− 1)x + y = n. Under the circumstances, the inequalityreduces to

x√

n−1y1√

n−1 [(n− 1)x2 + y2] ≤ n.

Forx = 0, the inequality is trivial. Forx > 0, it is equivalent tof(x) ≤ 0, where

f(x) =√

n− 1 ln x +1√

n− 1ln y + ln[(n− 1)x2 + y2]− ln n,

with y = n− (n− 1)x.




We havey′ = −(n− 1) and

f ′(x)√n− 1

=1

x− 1

y+

2√

n− 1(x− y)

(n− 1)x2 + y2=

(y − x)(√

n− 1x− y)2

xy[(n− 1)x2 + y2]≥ 0.

Therefore, the functionf(x) is strictly increasing on(0, 1] and hencef(x) ≤ f(1) = 0. Equal-ity occurs if and only ifx1 = x2 = · · · = xn = 1. �

Remark 3.13. Forn = 5, we get the following nice statement:If a, b, c, d, e are positive real numbers such thata2 + b2 + c2 + d2 + e2 = 5, then

abcde(a4 + b4 + c4 + d4 + e4) ≤ 5.

Proposition 3.14([4]). Let x, y, z be non-negative real numbers such thatxy + yz + zx = 3,and let

p ≥ ln 9− ln 4

ln 3≈ 0.738.

Then,xp + yp + zp ≥ 3.

Proof. Let r = ln 9−ln 4ln 3

. By the Power-Mean inequality, we have

xp + yp + zp

3≥(

xr + yr + zr

3

) pr

.

Thus, it suffices to show thatxr + yr + zr ≥ 3.

Let x ≤ y ≤ z. We consider two cases.

Casex = 0. We have to show thatyr + zr ≥ 3 for yz = 3. Indeed, by the AM-GM inequality,we get

yr + zr ≥ 2(yz)r/2 = 2 · 3r/2 = 3.

Casex > 0. The inequalityxr + yr + zr ≥ 3 is equivalent to the homogeneous inequality

xr + yr + zr ≥ 3(xyz

3

) r2

(1

x+

1

y+

1

z

) r2

.

Settingx = a1r , y = b

1r , z = c

1r (0 < a ≤ b ≤ c), the inequality becomes

a + b + c ≥ 3

(abc

3

) 12 (

a−1r + b

−1r + c

−1r

) r2.

Towards proving this inequality, we apply Corollary 1.9 (casep = 0, q = −1r

): If 0 < a ≤ b ≤ c

such thata + b + c = constant andabc = constant, then the suma−1r + b

−1r + c

−1r is maximal

when0 < a ≤ b = c.So, it suffices to prove the inequality for0 < a ≤ b = c; that is, to prove the homogeneous

inequality inx, y, z for 0 < x ≤ y = z. Without loss of generality, we may leave aside theconstraintxy + yz + zx = 3, and considery = z = 1 and0 < x ≤ 1. The inequality reduces to

xr + 2 ≥ 3

(2x + 1

3

) r2

.

Denoting

f(x) = lnxr + 2

3− r

2ln

2x + 1

3,



18 VASILE CÎRTOAJE

we have to show thatf(x) ≥ 0 for 0 < x ≤ 1. The derivative

f ′(x) =rxr−1

xr + 2− r

2x + 1=

r(x− 2x1−r + 1)

x1−r(xr + 2)(2x + 1)

has the same sign asg(x) = x − 2x1−r + 1. Sinceg′(x) = 1 − 2(1−r)xr , we see thatg′(x) < 0

for x ∈ (0, x1), andg′(x) > 0 for x ∈ (x1, 1], wherex1 = (2 − 2r)1/r ≈ 0.416. The functiong(x) is strictly decreasing on[0, x1], and strictly increasing on[x1, 1]. Sinceg(0) = 1 andg(1) = 0, there existsx2 ∈ (0, 1) such thatg(x2) = 0, g(x) > 0 for x ∈ [0, x2) andg(x) < 0for x ∈ (x2, 1). Consequently, the functionf(x) is strictly increasing on[0, x2] and strictlydecreasing on[x2, 1]. Sincef(0) = f(1) = 0, we havef(x) ≥ 0 for 0 < x ≤ 1, establishingthe desired result.

Equality occurs forx = y = z = 1. Additionally, for p = ln 9−ln 4ln 3

andx ≤ y ≤ z, equalityholds again forx = 0 andy = z =

√3. �

Proposition 3.15([7]). Letx, y, z be non-negative real numbers such thatx + y + z = 3, andlet p ≥ ln 9−ln 8

ln 3−ln 2≈ 0.29. Then,

xp + yp + zp ≥ xy + yz + zx.

Proof. Forp ≥ 1, by Jensen’s inequality we have

xp + yp + zp ≥ 3

(x + y + z

3

)p

= 3 =1

3(x + y + z)2 ≥ xy + yz + zx.

Assume nowp < 1. Let r = ln 9−ln 8ln 3−ln 2

andx ≤ y ≤ z. The inequality is equivalent to thehomogeneous inequality

2(xp + yp + zp)

(x + y + z

3

)2−p

+ x2 + y2 + z2 ≥ (x + y + z)2.

By Corollary 1.9 (case0 < p < 1 andq = 2), if x ≤ y ≤ z such thatx + y + z = constantandxp + yp + zp = constant, then the sumx2 + y2 + z2 is minimal when eitherx = 0 or0 < x ≤ y = z.

Casex = 0. Returning to our original inequality, we have to show thatyp + zp ≥ yz fory + z = 3. Indeed, by the AM-GM inequality, we get

yp + zp − yz ≥ 2(yz)p2 − yz

= (yz)p2 [2− (yz)

2−p2 ]

≥ (yz)p2

[2−

(y + z

2

)2−p]

= (yz)p2

[2−

(3

2

)2−p]

≥ (yz)p2

[2−

(3

2

)2−r]

= 0.




Case0 < x ≤ y = z. In the homogeneous inequality, we may leave aside the constraintx + y + z = 3, and considery = z = 1 and0 < x ≤ 1. Thus, the inequality reduces to

(xp + 2)

(x + 2

3

)2−p

≥ 2x + 1.

To prove this inequality, we consider the function

f(x) = ln(xp + 2) + (2− p) lnx + 2

3− ln(2x + 1).

We have to show thatf(x) ≥ 0 for 0 < x ≤ 1 andr ≤ p < 1. We have

f ′(x) =pxp−1

xp + 2+

2− p

x + 2− 2

2x + 1=

2g(x)

x1−p(xp + 2)(2x + 1),

whereg(x) = x2 + (2p− 1)x + p + 2(1− p)x2−p − (p + 2)x1−p,

andg′(x) = 2x + 2p− 1 + 2(1− p)(2− p)x1−p − (p + 2)(1− p)x−p,

g′′(x) = 2 + 2(1− p)2(2− p)x−p + p(p + 2)(1− p)x−p−1.

Sinceg′′(x) > 0, the first derivativeg′(x) is strictly increasing on(0, 1]. Taking into accountthatg′(0+) = −∞ andg′(1) = 3(1 − p) + 3p2 > 0, there isx1 ∈ (0, 1) such thatg′(x1) = 0,g′(x) < 0 for x ∈ (0, x1)andg′(x) > 0 for x ∈ (x1, 1]. Therefore, the functiong(x) is strictlydecreasing on[0, x1] and strictly increasing on[x1, 1]. Sinceg(0) = p > 0 andg(1) = 0, thereis x2 ∈ (0, x1) such thatg(x2) = 0, g(x) > 0 for x ∈ [0, x2) andg(x) < 0 for x ∈ (x2, 1]. Wehave alsof ′(x2) = 0, f ′(x) > 0 for x ∈ (0, x2) andf ′(x) < 0 for x ∈ (x2, 1]. According to thisresult, the functionf(x) is strictly increasing on[0, x2] and strictly decreasing on[x2, 1]. Since

f(0) = ln 2 + (2− p) ln2

3≥ ln 2 + (2− r) ln

2

3= 0

andf(1) = 0, we getf(x) ≥ min{f(0), f(1)} = 0.Equality occurs forx = y = z = 1. Additionally, for p = ln 9−ln 8

ln 3−ln 2andx ≤ y ≤ z, equality

holds again whenx = 0 andy = z = 32. �

Proposition 3.16([8]). If x1, x2, . . . , xn (n ≥ 4) are non-negative numbers such thatx1 +x2 +· · ·+ xn = n, then

1

n + 1− x2x3 · · ·xn

+1

n + 1− x3x4 · · ·x1

+ · · ·+ 1

n + 1− x1x2 · · ·xn−1

≤ 1.

Proof. Let x1 ≤ x2 ≤ · · · ≤ xn anden−1 =(1 + 1

n−1

)n−1. By the AM-GM inequality, we have

x2 · · ·xn ≤(

x2 + · · ·+ xn

n− 1

)n−1

≤(

x1 + x2 + · · ·+ xn

n− 1

)n−1

= en−1.

Hencen + 1− x2x3 · · ·xn ≥ n + 1− en−1 > 0,

and all denominators of the inequality are positive.

Casex1 = 0. It is easy to show that the inequality holds.

Casex1 > 0. Suppose thatx1x2 · · ·xn = (n + 1)r = constant,r > 0. The inequality becomesx1

x1 − r+

x2

x2 − r+ · · ·+ xn

xn − r≤ n + 1,



20 VASILE CÎRTOAJE

or1

x1 − r+

1

x2 − r+ · · ·+ 1

xn − r≤ 1

r.

By the AM-GM inequality, we have

(n + 1)r = x1x2 · · ·xn ≤(

x1 + x2 + · · ·+ xn

n

)n

= 1,

hencer ≤ 1n+1

. Fromxn < x1 + x2 + · · · + xn = n < n + 1 ≤ 1r, we getxn < 1

r. Therefore,

we haver < xi < 1r

for all numbersxi.We will apply now Corollary 1.7 to the functionf(u) = −1

u−r, u > r. We havef ′(u) = 1

(u−r)2

and

g(x) = f ′(

1

x

)=

x2

(1− rx)2, g′′(x) =

4rx + 2

(1− rx)4.

Sinceg′′(x) > 0, g(x) is strictly convex on(r, 1

r

). According to Corollary 1.7, if0 ≤ x1 ≤

x2 ≤ · · · ≤ xn such that forx1 + x2 + · · ·+ xn = constant andx1x2 · · ·xn = constant, then thesumf(x1) + f(x2) + · · · + f(xn) is minimal whenx1 ≤ x2 = x3 = · · · = xn. Thus, to provethe original inequality, it suffices to consider the casex1 = x andx2 = x3 = · · · = xn = y,where0 < x ≤ 1 ≤ y andx + (n− 1)y = n. We leave ending the proof to the reader. �

Remark 3.17. The inequality is a particular case of the following more general statement:Let n ≥ 3, en−1 =

(1 + 1

n−1

)n−1, kn = (n−1)en−1

n−en−1and leta1, a2, . . . , an be non-negative

numbers such thata1 + a2 + · · ·+ an = n.(a) If k ≥ kn, then

1

k − a2a2 · · · an

+1

k − a3a4 · · · a1

+ · · ·+ 1

k − a1a2 · · · an−1

≤ n

k − 1;

(b) If en−1 < k < kn, then1

k − a2a3 · · · an

+1

k − a3a4 · · · a1

+ · · ·+ 1

k − a1a2 · · · an−1

≤ n− 1

k+

1

k − en−1

.

Finally, we mention that many other applications of the EV-Method are given in the book [2].

REFERENCES

[1] V. CÎRTOAJE, A generalization of Jensen’s inequality,Gazeta Matematica A, 2 (2005), 124–138.

[2] V. CÎRTOAJE,Algebraic Inequalities - Old and New Methods, GIL Publishing House, Romania,2006.

[3] V. CÎRTOAJE, Two generalizations of Popoviciu’s Inequality,Crux Math., 31(5) (2005), 313–318.

[4] V. CÎRTOAJE, Solution to problem 2724 by Walther Janous,Crux Math., 30(1) (2004), 44–46.

[5] V. CÎRTOAJE, Problem 10528,A.M.M, 103(5) (1996), 428.

[6] V. CÎRTOAJE,Mathlinks site, http://www.mathlinks.ro/Forum/viewtopic.php?t=123604 , Dec 10, 2006.

[7] V. CÎRTOAJE,Mathlinks site, http://www.mathlinks.ro/Forum/viewtopic.php?t=56732 , Oct 19, 2005.

[8] V. CÎRTOAJE,Mathlinks site, http://www.mathlinks.ro/Forum/viewtopic.php?t=18627 , Jan 29, 2005.

[9] V. CÎRTOAJE,Mathlinks site, http://www.mathlinks.ro/Forum/viewtopic.php?t=14906 , Aug 04, 2004.


http://www.mathlinks.ro/Forum/viewtopic.php?t=123604










[10] V. CÎRTOAJE, Mathlinks site, http://www.mathlinks.ro/Forum/viewtopic.php?t=19076 , Nov 2, 2004.

[11] V. CÎRTOAJE, Mathlinks site, http://www.mathlinks.ro/Forum/viewtopic.php?t=21613 , Feb 15, 2005.

[12] W. JANOUSAND V. CÎRTOAJE, Two solutions to problem 2664,Crux Math., 29(5) (2003), 321–323.

[13] D.S. MITRINOVIC, J.E. PECARICAND A.M. FINK, Classical and New Inequalities in Analysis,Kluwer, 1993.

[14] J. SURANYI, Miklos Schweitzer Competition-Hungary.http://www.mathlinks.ro/Forum/viewtopic.php?t=14008 , Jul 11, 2004.

[15] http://www.mathlinks.ro/Forum/viewtopic.php?t=36346 , May 10, 2005.

[16] http://www.mathlinks.ro/Forum/viewtopic.php?p=360537 , Nov 2, 2005.









http://www.mathlinks.ro/Forum/viewtopic.php?p=360537


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Volume 8 (2007), Issue 1, Article 15, 21 pp.emis.maths.adelaide.edu.au/journals/JIPAM/images/059_06... · 2007-04-02 · Volume 8 (2007), Issue 1, Article 15, 21 pp. THE EQUAL VARIABLE